Primitive ideals in quantum Schubert cells: dimension of the strata

TL;DR

This paper investigates the structure of primitive ideals in quantum Schubert cells, providing explicit formulas for the dimensions of strata in their primitive spectrum, advancing understanding of their representation theory.

Contribution

It offers a new explicit formula for the dimension of strata associated with torus-invariant primes in quantum Schubert cells.

Findings

Stratification of primitive spectrum by torus-invariant primes

Explicit dimension formula for each stratum

Connection to maximal ideals of a torus

Abstract

The aim of this paper is to study the representation theory of quantum Schubert cells. Let $\g$ be a simple complex Lie algebra. To each element $w$ of the Weyl group $W$ of $\g$, De Concini, Kac and Procesi have attached a subalgebra $U_q[w]$ of the quantised enveloping algebra $U_q(\g)$. Recently, Yakimov showed that these algebras can be interpreted as the quantum Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of $U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter that the primitive spectrum of $U_q[w]$ admits a stratification indexed by those primes that are invariant under a natural torus action. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a given torus-invariant prime.